Talk:Milnor K-theory

"Milnor ring" (an alternative name) redirects here. BTotaro (talk) 17:40, 4 November 2015 (UTC)

Article improvements
There are quite a few useful results from Milnor's original paper "Algebraic K-theory and Quadratic forms" which should be included in this article for a better understanding of Milnor K-theory.

structure of K-theory ring

 * graded commutative ring structure
 * 1.2 and 1.3

computations

 * example 1.5 and implications
 * example 1.6 with generators, also include $$\mathbb{R}(t)$$ from section 2
 * $$K_n\mathbb{Q} \cong \mathbb{Z}/2$$ for $$ n \geq 3$$ can be deduced from later methods

structure results

 * example 1.7 gives partial computation of local fields : they are all divisible, moreover, using the same argument as 1.5 this gives all milnor K-groups

theorems

 * theorem 2.3 (give exact sequence for Q(t), R(t), C(t) (or any algebraically closed field),  showing the structure)
 * C(t_1), C(t_1)(t_2), C(t_1)(t_2)(t_3), ...
 * lemma 6.2 -> relation with Galois cohomology (refined further later on Bloch-Kato)
 * A.2

Applications section

 * theorem 1.4 for arithmetic

Other articles
Wundzer (talk) 17:15, 22 January 2021 (UTC)
 * Also, mention Voevodsky's article on motivic cohomology (corollary 7.5 on page 97) https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/motivic_cohomology_with_Z2_coefficients_published.pdf
 * Give motivic sheaves representing Milnor K-theory https://arxiv.org/abs/math/0107109
 * Relate to motivic cohomology, higher chow groups, and higher algebraic K-theory, this shows Milnor K-theory is part of higher algebraic K-theory

Higher local class field theory
This article should mention the main theorem of local class field theory. The statement can be found in and contains other useful pdfs. Also, contains useful stuff on Milnor K-theory, starting on page 292 of the pdf. Kaptain-k-theory (talk) 20:40, 14 April 2021 (UTC)
 * https://arxiv.org/abs/math/0012136
 * https://msp.org/gtm/2000/03/
 * https://www.maths.nottingham.ac.uk/plp/pmzibf/book/vol.pdf
 * Differential forms and Milnor K-theory

Motivic steenrod algebra
Should discuss the relation between motivic cohomology and Milnor K-theory. In addition, discuss the various results about the motivic steenrod algebra and motivic eilenberg-maclane spaces. Some resources include
 * http://www.mathematik.ur.de/hoyois/papers/motivicsteenrod.pdf
 * https://people.math.rochester.edu/faculty/doug/otherpapers/powell.pdf
 * https://conf.math.illinois.edu/K-theory/0170/milnor.pdf

Kaptain-k-theory (talk) 18:19, 28 April 2021 (UTC)